Abstract
We present a top-down lower bound method for depth-three ⋎, ⋏, ¬-circuits which is simpler than the previous methods and in some cases gives better lower bounds. In particular, we prove that depth-three ⋎, ⋏, ¬-circuits that compute parity (or majority) require size at least\(2^{0.618...\sqrt n } (or 2^{0.849...\sqrt n } \), respectively). This is the first simple proof of a strong lower bound by a top-down argument for non-monotone circuits.
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References
M. Ajtai, 111-1 on finite structures.Ann. Pure and Appl. Logic 24 (1983), 1–48.
A. Borodin, A. Razborov andR. Smolensky, On lower bounds for reak-k-times branching programs.Computational Complexity 3 (1993), 1–18.
P. Erdös andR. Rado, Intersection theorems for systems of sets.J. London Math. Soc. 35 (1960), 85–90.
M. Furst, J. Saxe andM. Sipser, Parity, circuits and the polynomial time hierarchy.Math. Systems Theory 17 (1984), 13–27.
J. Håstad,Almost Optimal Lower Bounds for Small Depth Circuits. Advances in Computing Research, ed.S. Micali, Vol 5 (1989), 143–170.
S. Jukna,Finite limits and lower bounds for circuit size. Tech. Rep. 94-06, Informatik, University of Trier, 1994.
M. Karchmer andA. Wigderson, Monotone circuits for connectivity require super-logarithmic depth.SIAM J. Disc. Math. 3 (1990), 255–265.
M. Klawe, W. J. Paul, N. Pippenger, M. Yannakakis, On monotone formulae with restricted depth. InProc. Sixteenth Ann. ACM Symp. Theor. Comput., 1984, 480–487.
R. Raz and A. Wigderson, Monotone circuits for matching require linear depth. InProc. Twenty-second Ann. ACM Symp. Theor. Comput., 1990, 287–292.
A. A. Razborov, Lower bounds for the size of circuits of bounded depth with basis {⋏, ⊕}.Math. Notes of the Academy of Sciences of the USSR 41:4 (1987), 333–338.
M. Sipser, Private communication, 1991.
M. Sipser, A topological view of some problems in complexity theory. InColloq. math. Soc. János Bolyai 44 (1985), 387–391.
R. Smolensky, Algebraic methods in the theory of lower bounds for Boolean circuit complexity. InProc. Nineteenth Ann. ACM Symp. Theor. Comput., 1987, 77–82.
L.G. Valiant, Graph-theoretic arguments in low level complexity. InProc. Sixth Conf. Math. Foundations of Computer Science, Lecture Notes in Computer Science, 1977, Springer-Verlag, 162–176.
A.C. Yao, Separating the polynomial time hierarchy by oracles. InProc. Twentysixth Ann. IEEE Symp. Found. Comput. Sci., 1985, 1–10.
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Håstad, J., Jukna, S. & Pudlák, P. Top-down lower bounds for depth-three circuits. Comput Complexity 5, 99–112 (1995). https://doi.org/10.1007/BF01268140
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DOI: https://doi.org/10.1007/BF01268140